Find the average rate of change for f(x)=x^2+2x+5 over the interval [-4,a]

To find the **average rate of change** of the function \( f(x) = x^2 + 2x + 5 \) over the interval \([-4, a]\), we use the formula for the average rate of change between two points \(x_1\) and \(x_2\) on a function \(f(x)\):

\[
\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]

Here:
- \(x_1 = -4\)
- \(x_2 = a\)

The formula becomes:

\[
\text{Average Rate of Change} = \frac{f(a) - f(-4)}{a - (-4)} = \frac{f(a) - f(-4)}{a + 4}
\]

### Step 1: Calculate \( f(-4) \)

\[
f(-4) = (-4)^2 + 2(-4) + 5 = 16 - 8 + 5 = 13
\]

### Step 2: Calculate \( f(a) \)

\[
f(a) = a^2 + 2a + 5
\]

### Step 3: Plug in the values into the formula

\[
\text{Average Rate of Change} = \frac{a^2 + 2a + 5 - 13}{a + 4} = \frac{a^2 + 2a - 8}{a + 4}
\]

So, the average rate of change of the function \( f(x) = x^2 + 2x + 5 \) over the interval \([-4, a]\) is:

\[
\boxed{\frac{a^2 + 2a - 8}{a + 4}}
\]

Do you want any further details or explanations?

Here are 5 related questions:
1. How do you calculate the rate of change for linear functions?
2. What happens to the average rate of change as \( a \) increases?
3. How does the function's derivative relate to its average rate of change?
4. Can we simplify the expression further for specific values of \( a \)?
5. What is the interpretation of the average rate of change for non-linear functions?

**Tip:** The average rate of change over an interval gives the slope of the secant line through the points on the curve at the interval endpoints.

Learn how to find the average rate of change of the quadratic function f(x) = x^2 + 2x + 5 over the interval [-4, a]. This solution covers key algebra concepts and step-by-step calculations, including evaluating the function at specific points. Suitable for students in Grades 9-12.

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